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# Units

(Redirected from Unit)

An element of a set is a unit if its multiplicative inverse belongs to that set. (The set of units form a multiplicative group.) Algebraic integer units have complex norm
 1
and are thus complex roots of unity.

## Units in quadratic integer rings

Imaginary quadratic integer rings have only a few units (see Theorem I1). The primitive root of unity of degree
 3
${\displaystyle \omega :=e^{i{\tfrac {2\pi }{3}}}={\frac {-1}{2}}+{\frac {\sqrt {-3}}{2}},}$

is used in the following table.

Units of imaginary quadratic integer rings
 d
Units
@opsp@@opsp@≤@opsp@@opsp@ @opsp@−@opsp@5 ${\displaystyle \{(-1)^{0},(-1)^{1}\}=\{1,-1\}}$
@opsp@−@opsp@3 ${\displaystyle \{\omega ,-\,\omega ,\omega ^{2},-\,\omega ^{2},1,-1\}}$
@opsp@−@opsp@2 ${\displaystyle \{(-1)^{0},(-1)^{1}\}=\{1,-1\}}$
@opsp@−@opsp@1 ${\displaystyle \{i^{0},i^{1},i^{2},i^{3}\}=\{1,i,-1,-i\}}$

Real quadratic integer rings have infinitely many units (see Theorem R1), which are all powers of each other, some multiplied by
 1
. For that reason, the table below can't be complete like the table above. An inefficient way to find a unit of a real quadratic integer ring
 ℤ
 [2√  d  ]
other than
 1
or
 1
is to try each positive value of
 b
starting with
 1
and going up until
 2√  1 + d b 2
is an integer.

Units of real quadratic integer rings
 d
Units
2 ${\displaystyle 1+{\sqrt {2}}}$
3 ${\displaystyle 2+{\sqrt {3}}}$
4 N/A
5 ${\displaystyle {\frac {1}{2}}+{\frac {\sqrt {5}}{2}}}$ (golden ratio)
6 ${\displaystyle 5+2{\sqrt {6}}}$
7 ${\displaystyle 8+3{\sqrt {7}}}$
8 N/A, but note that ${\displaystyle 3^{2}-8(1^{2})=1}$
9 N/A
10 ${\displaystyle 3+{\sqrt {10}}}$
11 ${\displaystyle 10+3{\sqrt {11}}}$
12 N/A, but ${\displaystyle 7^{2}-12(2^{2})=1}$
13 ${\displaystyle {\frac {3}{2}}+{\frac {\sqrt {13}}{2}}}$
14 ${\displaystyle 15+4{\sqrt {14}}}$
15 ${\displaystyle 4+{\sqrt {15}}}$
16 N/A
17 ${\displaystyle 4+{\sqrt {17}}}$
18 N/A, but ${\displaystyle 17^{2}-18(4^{2})=1}$
19 ${\displaystyle 170+39{\sqrt {19}}}$
20 N/A, but ${\displaystyle 9^{2}-20(2^{2})=1}$
21 ${\displaystyle {\frac {5}{2}}+{\frac {\sqrt {21}}{2}}}$
22 ${\displaystyle 197+42{\sqrt {22}}}$
23 ${\displaystyle 24+5{\sqrt {23}}}$
24 N/A, but ${\displaystyle 5^{2}-24(1^{2})=1}$
25 N/A
26 ${\displaystyle 5+{\sqrt {26}}}$
27 N/A, but ${\displaystyle 26^{2}-27(5^{2})=1}$
28 N/A, but ${\displaystyle 127^{2}-28(24^{2})=1}$
29 ${\displaystyle {\frac {5}{2}}+{\frac {\sqrt {29}}{2}}}$
30 ${\displaystyle 11+2{\sqrt {30}}}$
31 ${\displaystyle 1520+273{\sqrt {31}}}$
32 N/A, but ${\displaystyle 17^{2}-32(3^{2})=1}$
33 ${\displaystyle {\frac {23}{2}}+{\frac {4{\sqrt {33}}}{2}}}$

## Examples

• The units of  ℕ
are  {1 0} = {1}
, where  1   ≡   e i 2π
. (See natural numbers.)
• The units of  ℤ
are  {( − 1) 0, ( − 1) 1} = {1,  − 1}
, where  − 1   ≡   e i π
. (See rational integers.)
• The units of  ℤ [ω]
are  {ω 0, ω 1, ω 2} = {1, ω, ω 2}
, where
ω   ≡   ei 2π 3
. (See Eisenstein integers.)
• The units of  ℤ [i]
are  {i 0, i 1, i 2, i 3} = {1, i,  − 1,  −  i}
, where
i   ≡   ei π 2
. (See Gaussian integers.)
• The units of  ℚ
are  ℚ \ {0}
.
• The units of  ℝ
are  ℝ \ {0}
.
• The units of  ℂ
are  ℂ \ {0}
.